Activities that let students get immediate feedback on how their are doing are extremely beneficial. Activities that allow students to self-check their own work allow for this immediate feedback and correction. These types of activities can allow the teacher to focus their time with students who are having conceptual misunderstandings and not get bogged down helping students find and correct computational errors. While students are engaged in self-checking activities, the teacher can also be working with small groups of students on mini-lessons or conversations/conferences. Below are a few of my favourite activities and routines that allow for students to check their own work:

Team Post-Its - I recently saw this activity described in a post by Julie Morgan. This activity is very easy to set up and does not require much front loaded time to create. The teacher posts a list of questions for small groups of students to work on. These might even be questions from the textbook. Each group solves the question, writes their answer on a sticky note and posts it on the whiteboard. As other groups complete the questions, they can compare their answers to those from other groups to see if they agree. If they don't agree, they double check their work. I would suggest that each group of students starts with a different question.

​Add ‘Em Up - In this activity, students do a set of problems, either on their own or as a group. These problems typically have numerical answers. The answers to the set of problems are added up and compared to the sum provided. If the sum is not the same, then the student knows that one or more of the problems in the set was done incorrectly and works to find the error. I first saw this activity described in a blog post from Kate Nowak. I later saw a blog post from Amy Gruen describing a simple and quick way to do this same activity that I used occasionally. There are many descriptions of how to organize this activity including one in a detailed blog post from Sara VanDerWerf.

​Row Game - I also first saw this activity described in a blog post from Kate Nowak. Typically, a row game is a worksheet of problems organized in two columns. The worksheet is completed by a pair of students, one doing the problems in column A and the other doing the problems in column B. The problems in each row have the same answer so if the students' answers don't match, they can work together to check their solutions to find the error. To make row games a bit easier to create, you can create an additional column with the sum of the solutions from column A and B (similar to the Add 'Em Up activity from above). This allows you to use any two problems and not have to create two problems with the same solution. Kate Nowak has a shared google folder with a large selection of crowd-sourced row games.

Added 03Oct017 - I recently saw a great idea from Heidi Neufeld. She asked students who finished quickly to make a new row for the row game and create two different problems with the same answer.

Tarsia Puzzles - You can download a free program that makes creating these puzzles very straight forward. Print them off on coloured cardstock and have students cut them out and then try to assemble them into a given shape. If they can't make the shape then they know that there is an error to find.

Mathematical Circuit Training / Around the World / Star Chain / Question Stack - There are lots of different names for and variations of this activity. The essential part is that there is a series of problems and the answer for each problem leads you to the next question to ask. The answer to the final question leads you back to the starting question. This activity can be organized as a simple worksheet, a stack of cards to turn over, a set of cards to chain together or questions posted on signs around the classroom or hallway. If you make a mistake, you won't be able to find the next question and you know to try again. This can be done individually or in small groups.

Added 12Oct2017 - I recently saw this same activity shared on twitter from Emily Libbert as a game for a math station. Expressions and solutions are written on popsicle sticks that must be placed in order.

Added 26Sep2017 - Thanks Alicia!Invisible Ink - The description of this activity is from a blog post from David Petro. Students solve a set of problems on a card. When ready, they can check their solutions using an answer card. This card has the correct answers written with "invisible" ink that can only be seen by shining a small UV light on it. Once the student has solve the questions correctly, they move on to the next card containing more complex questions. David says, "​Students really seem to like this style of activity as they feel empowered to move from card to card when they are ready and the added feature of checking the answers with the UV pen gives a sense of novelty."

If you know of any other self-checking activities that I've missed, please let me know and I'll add them here.

I recently wrote an article about podcasts for the Global Math Department newsletter. I had more to write about podcasts, so I'm continuing my thoughts here. There are lots of really well done podcasts focused on mathematics and education and I wanted to share the best of what I've been listening to on my daily commute to and from work. I have just enough of a commute to get through about a podcast each day. I really enjoy how a good podcast make a dull commute alone in a car feel like a road trip with friends.

One new podcast that I started listening to is called Infinite Insights. Duane Habecker and Maggie Peters started this podcast to discuss articles and research regarding mathematics education and how it applies to your classroom. I really enjoyed listening to their first podcast about using math for your Day 1 routines.

Five Fabulous Podcasts Episodes

Instead of just recommending a bunch of podcasts, I wanted to hone in on specific episodes that I think you'll enjoy (assuming you're into mathematics). Check these out:

On Average from 99 Percent Invisible. ﻿Roman Mar﻿s discusses how the the built world was not designed for you but for the average person.

For the Love of Numbers from Radiolab. In this episode, ﻿Alex Bellos﻿ recalls being asked about his favourite number. He talks about the genesis of his interest in favourite numbers and how people throughout history have ascribed human-like qualities to numbers.

Dave Coffey & John Golden from Math Ed Podcast. Samuel Otten talks with a pair of his former professors about how they use blogs and Twitter in their professional work as mathematics educators.

So what makes a good podcast? I used four criteria for selecting the podcasts above: Length (I find that 30-40 minutes is about right for me), Mathematical Interest (I like to learn new things about math or education), Content Density (I like a podcast to get to its point and not linger on idle chit chat), and Production Value (a good microphone and a sound booth help).

I'd really like to hear about your favourite mathematics or education podcasts. Even better, I'd like to hear about a great episode of your favourte podcast. Please let me know what I've been missing out on.

There seems to be a lot of discussion about "what makes a good question." There are entire books filled with examples of rich question and engaging problems. Two of my favourites are Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small and Good Questions for Math Teaching: Why Ask Them and What to Ask, Grades 5-8 by Lainie Schuster and Nancy L. Anderson. I've seen much less discussion about "what makes a question good." I think it's harder to define the effective teaching practices and routines for asking questions in a manner that makes them effective.

Good Questions

What are the characteristics of a good question? I recently read a post from Geoff Krall in which he wrote about teachers adapting questions they had found to make them even better. He said that to do this, "You start to turn from 'I like this task' to 'What do you like about it?'" I think that this is an important question to ask yourself. Teachers need to think about their students and the context in which they teach to determine what questions are going to best facilitate effective student learning in their classroom. Here are some possible characteristics of good questions that you might consider:

They are aligned with the outcomes and goals for the lesson that I'm teaching.

Rich Classrooms

Jennifer Piggott in an article on NRICH wrote, "In essence, rich tasks encourage children to think creatively, work logically, communicate ideas, synthesise their results, analyse different viewpoints, look for commonalities and evaluate findings. However, what we really need are rich classrooms: communities of enquiry and collaboration, promoting communication and imagination." This really resonated with me. It is not enough to have a great problem. As teachers we need to know how to effectively present and lead the exploration of a problem in order to reap its benefits.

There are a number of strategies you might consider when exploring a question:

Introduce the problem in a way that allows students to understand the problem and promotes student ownership. (e.g. a Notice and Wonder routine, modeling the problem, including a personal anecdote, etc.)

Allow students to working collaboratively (at least part of the time) and encourage discussion. Groups of students might record their work and share their conclusions using white boards or other non-permanent surfaces.

You Need Both to Succeed

My grandmother was a big fan of the card game pinochle and she taught me to play at an early age. It's a trick taking card game where players work in pairs to score points. I learned early that even the best hand of cards can be beaten by someone who knows how to play well. My grandmother played to win and she would often remind us to "mind your p's and q's" while playing to pay attention and play carefully. I learned a lot of math counting up tricks and keeping score. The important lesson here is that to be successful at pinochle, you have to have both a good hand of cards to meld and play those cards correctly to maximize your score. Just like pinochle, when planning a lesson, you need to consider two important and complementary components. Both finding good questions and using them effectively are equally important to the success of your problem solving lesson.